ACTUARY Mathematics and Statistics Flashcards

A chi-square goodness-of-fit test is used to determine whether observed frequencies match expected frequencies. The test statistic follows a chi-square distribution with how many degrees of freedom if there are k categories?

Answer: k-1

With k categories and no estimated parameters, the degrees of freedom equal k-1.

If X follows a lognormal distribution with parameters μ and σ², what is E[X]?

Answer: e^(μ + σ²/2)

For a lognormal distribution, E[X] = e^(μ + σ²/2).

An insurer observes 200 losses from a policy. The maximum likelihood estimator for the mean of an exponential distribution is:

Answer: The sample mean X̄

For the exponential distribution, the MLE of the mean θ is the sample mean X̄.

Which of the following statements about the F-distribution is correct?

Answer: It is the ratio of two independent chi-square variables each divided by their degrees of freedom

The F-distribution is defined as the ratio of two independent chi-square random variables, each divided by its degrees of freedom.

A compound Poisson process has claim frequency N ~ Poisson(λ) and i.i.d. claim severities X_i. What is the variance of the aggregate loss S = ΣXᵢ?

Answer: λ·E[X²]

For a compound Poisson process, Var(S) = λ·E[X²], which equals λ(Var(X) + (E[X])²).

The Kolmogorov-Smirnov test compares:

Answer: The empirical CDF to a theoretical CDF

The KS test measures the maximum absolute difference between the empirical CDF and a specified theoretical CDF.

For a negative binomial distribution with parameters r and p, the mean is r(1-p)/p and the variance is r(1-p)/p². Compared to the Poisson, the negative binomial is said to exhibit:

Answer: Overdispersion

Since Var > Mean for the negative binomial, it exhibits overdispersion, making it useful for modeling claim counts with high variability.